%\addcontentsline{toc}{chapter}{Notation}
\chapter{Notation} \label{appendix:notation}
%
%\fancyhead[LO]{\textsc{Notation}}
%\fancyhead[RE]{\textsc{Notation}}
%
\lettrine[lines=3]{T}{his}
appendix lists symbols that appear often throughout this thesis.
The symbols are grouped together according to relevance and topic.
For each symbol, we present a short description and a reference to the page
where that symbol is introduced and defined for the first time.
\cbstart
For the general mathematical notation used throughout this thesis, no
page references are given.
\cbend

\section*{Mathematics}
\begin{tabbing}
$\mathbb{N}$~~~~~~~~~~~~\=Set of natural numbers, $0, 1, 2, \ldots$\\
\gensym \mathbb{R} : {Set of real numbers}
\gensym \mathbb{R}^n : {Set of vectors of $n$ real numbers, $n > 0$}
\gensym [a,b] : {Closed interval between the real numbers $a$ and $b$}
\gensym [a,b) : {Half-open interval between the real numbers $a$ and $b$}
\gensym {\lfloor\,\cdot\,\rfloor} : {Floor of a real number}
\gensym {\lceil\,\cdot\,\rceil} : {Ceiling of a real number}
\gensym \transpose{A} : {The transpose of the matrix (or vector) $A$}
\gensym \expectedvalue{\cdot} : {Expected value of a stochastic variable}
\gensym \stddev{\cdot} : {Standard deviation of a stochastic variable}
\gensym \algAssignment : {Assignment or initialization}
%\\
$f : X \longrightarrow Y\;$ Function from the set $X$ to the set $Y$\\
$f^{-1} : Y \longrightarrow X\;$ Inverse of the function $f$\\
\gensym \times : {Cartesian product of sets}
\gensym {\lvert\,\cdot\,\rvert} : {Cardinality of a finite set}
\gensym \cup : {Union of sets}
\gensym \cap : {Intersection of sets}
\gensym \setminus : {Set difference}
\gensym \subset : {Subset relation}
\gensym \subseteq : {Subset-or-equality relation}
\gensym \emptyset : {The empty set}
\gensym \powerset{X} : {Power set of the set $X$}
\gensym {\{a_i\}_{i\in\indexset}} : {Set of items indexed by $\indexset$}
\end{tabbing}



\section*{Control Applications}
\begin{tabbing}
% YOU NEED TO ADD THE FIRST ONE MANUALLY TO ADJUST THE TABBING AND SPACES
%$first$~~~~~~~~~~~~\=First symbol, page 0\\
$\plants$~~~~~~\=Set of plants, page \pageref{symbol:plantset}\\
%ADD THE REST OF SYMBOLS WITH THE HELP OF MACRO
%\addsymbol \plants : {Set of plants}{symbol:plantset}
\addsymbol \indexset_\plants : {Index set of $\plants$}{symbol:indexsetplants}
\addsymbol \plant_i : {Plant ($i \in \indexset_\plants$)}{symbol:planti}
\addsymbol \plantstate_i: {Plant state}{symbol:plantstatei}
\addsymbol \plantinput_i: {Plant input (actuators)}{symbol:plantinputi}
\addsymbol \plantoutput_i: {Plant output (sensors)}{symbol:plantoutputi}
\addsymbol \applicationset : {Set of control applications}{symbol:applicationset}
\addsymbol \application_i : {Control application ($i \in \indexset_\plants$)}{symbol:applicationi}
\addsymbol \taskset_i : {Set of tasks in control application $\application_i$}{symbol:taskseti}
\addsymbol \indexset_i : {Index set of $\taskset_i$}{symbol:indexsettaskseti}
\addsymbol \msgset_i : {Set of messages of control application $\application_i$}{symbol:msgseti}
\addsymbol \taskset_\applicationset : {Set of all tasks $\bigcup_{i \in \indexset_\plants} \taskset_i$}{symbol:alltasks}
\addsymbol \task : {Task}{symbol:task}
\addsymbol \task_{ij} : {Arbitrary task of application $\application_i$ ($i \in \indexset_\plants$, $j\in\indexset_i$)}{symbol:taskij}
\addsymbol \job{ij}{q} : {Job $q$ of task $\task_{ij}$}{symbol:jobij}
\addsymbol \msg_{ijk} : {Message between tasks $\task_{ij}$ and $\task_{ik}$ in application $\application_i$}{symbol:msgijk}
\addsymbol \msginstance{ijk}{q} : {Instance $q$ of message $\msg_{ijk}$}{symbol:msginstijk}
\addsymbol \controlcost_i: {Control cost for plant $\plant_i$ (quality metric)}{symbol:controlcosti}
\end{tabbing}


\section*{Platform}
\begin{tabbing}
$\nodeset$~~~~~~~~~~~~\=Set of computation nodes, page \pageref{symbol:nodeset}\\
%\addsymbol \nodeset : {Set of computation nodes}{symbol:nodeset}
\addsymbol \indexset_\nodeset : {Index set of $\nodeset$}{symbol:indexsetnodeset}
\addsymbol \mapping : {Task mapping to computation nodes}{symbol:mapping}
\addsymbol \mappinglimitation : {Specification of allowed nodes for each task}{symbol:mappinglimitation} 
\addsymbol \maxmemory_d : {Memory on node $\node_d$ ($d \in \indexset_\nodeset$) for design solutions}{symbol:maxmemd}
\end{tabbing}


\section*{Control Timing}
\begin{tabbing}
$\period_i$~~~~~~\=Period of control application $\application_i$, page \pageref{symbol:periodi}\\
%\addsymbol \period_i : {Period of control application $\application_i$}{symbol:periodi}
\addsymbol \hyperperiod : {Hyper period of the system}{symbol:hyperperiod}
\addsymbol \periodset_i : {Set of allowed periods of application $\application_i$}{symbol:periodset}
\addsymbol \boldsymbol{\period} : {Period assignment of the application set $\applicationset$}{symbol:periodassignment}
\addsymbol \bcet{ij} : {Best-case execution time of $\task_{ij}$}{symbol:bcetij}
\addsymbol \wcet{ij} : {Worst-case execution time of $\task_{ij}$}{symbol:wcetij}
\addsymbol \pdf_{\exectime_{ij}} : {Probability function of the execution time of $\task_{ij}$}{symbol:pdfexectimeij}
\addsymbol \exectime_{ijk} : {Communication time of $\msg_{ijk}$ on the bus}{symbol:commtimeijk}
\addsymbol \delay_i^\textrm{sa} : {Sensor--actuator delay}{symbol:SAdelay}
\addsymbol \DelaySA_i : {Stochastic variable representing sensor--actuator delay}{symbol:stochdelaySA}
\addsymbol \DelaySApdf_{\Delay_i} : {Probability function of $\DelaySA_i$}{symbol:pdfdelaySA}
\end{tabbing}



\section*{Multi-Mode Systems}
\begin{tabbing}
$\modes$~~~~~~~~~~~~\=Set of modes, page \pageref{symbol:modeset}\\
\addsymbol \functionalmodes : {Set of functional modes}{symbol:functionalmodes}
\addsymbol \virtualmodes : {Set of virtual modes}{symbol:virtualmodes}
\addsymbol \mode : {Mode}{symbol:mode}
\addsymbol \indexset_\mode : {Index set of mode $\mode$}{symbol:indexsetmode}
\addsymbol \supermodes{\mode} : {Supermodes of $\mode$}{symbol:supermodes}
\addsymbol \submodes{\mode} : {Submodes of $\mode$}{symbol:submodes}
\addsymbol \topmodes{\modes^\prime} : {Top modes of the set of modes $\modes^\prime \subseteq \modes$}{symbol:topmodes}
\addsymbol \implementedmodes : {Set of implemented modes}{symbol:implementedmodes}
\addsymbol \modecost{\mode} : {Total control cost in mode $\mode$}{symbol:overallmodecost}
\addsymbol \controlcost_i^\mode : {Control cost of $\application_i$ in mode $\mode$}{symbol:modecosti}
\addsymbol \modecostwithsupermode{\mode}{\mode^\prime} : {Control cost of $\mode$ with the solution of $\mode^\prime$}{symbol:modecostwithsupermode}
\addsymbol \modememreq{\mode}{d} : {Memory space on $\node_d$ for the solution of $\mode$}{symbol:memreq}
\end{tabbing}


\section*{Fault-Tolerant Systems}
\begin{tabbing}
$\configurationset$~~~~~~~~~~~~~~~\=Set of configurations, page \pageref{symbol:configurationset}\\
\addsymbol \nodeconfig : {Configuration}{symbol:nodeconfig}
\addsymbol \feasibleconfigset : {Set of feasible configurations}{symbol:feasibleconfigset}
\addsymbol \baseconfigset : {Set of base configurations}{symbol:baseconfigset}
%\addsymbol \minimumavailablenodes : {Minimum number of available nodes}{symbol:minavailnodes}
\addsymbol \minconfigset : {Set of minimal configurations}{symbol:minconfigset}
\addsymbol \opconfigset : {Set of configurations with resilience to faults}{symbol:opconfigset}
\addsymbol \mapping_\nodeconfig : {Task mapping to configuration $\nodeconfig$}{symbol:configmapping}
\addsymbol \migrationtime(\task) : {Migration time of task $\task$}{symbol:taskmigrationtime}
\addsymbol \migrationtimemax : {Maximum allowed migration time}{symbol:migrationtimemax}
\addsymbol \mem_d^\nodeconfig : {Memory space on $\node_d$ for the solution of $\nodeconfig$}{symbol:nodememconfig}
\addsymbol \mem_d(\task) : {Memory space of $\task$ on $\node_d$}{symbol:memtasknode}
\addsymbol \mappedconfigset : {Set of synthesized optional configurations}{symbol:mappedconfigset}
\addsymbol \controlcost^\nodeconfig : {Overall control cost in configuration $\nodeconfig$}{symbol:overallconfigcost}
\addsymbol \failureprob(\node) : {Failure probability of node $\node \in \nodeset$}{symbol:failureprob}
\addsymbol \failureprob^\nodeconfig : {Probability of reaching configuration $\nodeconfig$}{symbol:failureprobconfig} 
\addsymbol \storedtasks{d} : {Set of tasks that are stored on node $\node_d$ ($d \in \indexset_\nodeset$)}{symbol:storedtasks}
\end{tabbing}


\section*{Self-Triggered Control}
\begin{tabbing}
$\taskset$~~~~~~~~\=Set of self-triggered control tasks, page \pageref{symbol:taskset}\\
\addsymbol \indexset_\taskset : {Index set of $\taskset$}{symbol:indexsettaskset}
\addsymbol \deadline_i : {Absolute deadline of the next execution of $\task_{i}$}{symbol:absdeadlinei}
\addsymbol \reldeadline_i : {Relative deadline of the next execution of $\task_{i}$}{symbol:reldeadlinei}
\addsymbol \minreldeadline_i : {Minimum relative deadline of $\task_i$}{symbol:minreldeadline}
\addsymbol \maxreldeadline_i : {Maximum relative deadline of $\task_i$}{symbol:maxreldeadline}
\addsymbol \currenttime_i : {Completion time of the latest execution of $\task_i$}{symbol:completiontimei}
\addsymbol \starttime_i : {Start time of the next execution of $\task_i$}{symbol:starttimei}
\addsymbol \statecost_{i} : {State cost of $\task_{i}$ in the time interval $[\currenttime_{i},\deadline_{i}]$}{symbol:statecosti}
\addsymbol \normalizedstatecost_i : {Control cost of $\task_i$ in the time interval $[\currenttime_{i},\deadline_{i}]$}{symbol:selftriggeredcontrolcost}
\addsymbol \cpucost_i : {CPU cost of $\task_i$ in the time interval $[\currenttime_{i},\deadline_{i}]$}{symbol:CPUcost}
\addsymbol \cost_i : {Combined control and CPU cost of $\task_i$}{symbol:selftriggeredoverallcost}
\addsymbol \costtradeoff : {Trade-off between control and CPU cost}{symbol:costtradeoff}
\addsymbol \statecostapprox_i : {Approximate state cost of $\task_{i}$}{symbol:approxstatecost}
\addsymbol \normalizedstatecostapprox_i : {Approximate control cost of $\task_{i}$}{symbol:approximatecontrolcost}
\addsymbol \approximation{\cost}_i : {Approximate combined control and CPU cost of $\task_{i}$}{symbol:approximateoverallcost}
\addsymbol \candidatestarttimes_i : {Set of candidate start times of task $\task_i$}{symbol:candidatestarttimes}
\addsymbol \schedulablecandidatestarttimes_i : {Set of schedulable candidate start times of task $\task_i$}{symbol:schedulablecandidatestarttimes}
\end{tabbing}












